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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/>







<b>thus </b><a NAME="E1:51"/>
( <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> implies (  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:51_1"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<b>thus </b><a NAME="E3:51_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E4:51_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:51_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E6:51_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:51_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> ;<br/><a NAME="E3:51_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:51_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E5:51_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
<font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


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